Foundations of Quantitative Finance Book V General Measure and Integration Theory 1St Edition Robert R Reitano Full Chapter
Foundations of Quantitative Finance Book V General Measure and Integration Theory 1St Edition Robert R Reitano Full Chapter
Foundations of Quantitative Finance Book V General Measure and Integration Theory 1St Edition Robert R Reitano Full Chapter
Dilip B. Madan
Robert H. Smith School of Business
University of Maryland, USA
Rama Cont
Mathematical Institute
University of Oxford, UK
Robert A. Jarrow
Ronald P. & Susan E. Lynch Professor of Investment ManagementSamuel Curtis Johnson Graduate School of
Management Cornell University
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DOI: 10.1201/9781003264576
Preface xi
Author xiii
Introduction xv
1 Measure Spaces 1
1.1 Lebesgue and Borel Spaces on R . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 Starting Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.2 Lebesgue Measure Space . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Borel Measure Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 General Extension Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Measure Space Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Finite Products of Measure Spaces . . . . . . . . . . . . . . . . . . . 11
1.3.2 Borel Measures on Rn . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.3 Infinite Products of Probability Spaces . . . . . . . . . . . . . . . . . 14
1.4 Continuity of Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Measurable Functions 17
2.1 Properties of Measurable Functions . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Limits of Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Results on Function Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Approximating σ(X)-Measurable Functions . . . . . . . . . . . . . . . . . . 26
2.5 Monotone Class Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.1 Monotone Class Theorem . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5.2 Functional Monotone Class Theorem . . . . . . . . . . . . . . . . . . 34
vii
viii Contents
4 Change of Variables 84
4.1 Change of Measure: A Special Case . . . . . . . . . . . . . . . . . . . . . . 85
4.1.1 Measures Defined by Integrals . . . . . . . . . . . . . . . . . . . . . . 85
4.1.2 Integrals and Change of Measure . . . . . . . . . . . . . . . . . . . . 89
4.2 Transformations and Change of Measure . . . . . . . . . . . . . . . . . . . 92
4.2.1 Measures Induced by Transformations . . . . . . . . . . . . . . . . . 92
4.2.2 Change of Variables under Transformations . . . . . . . . . . . . . . 95
4.3 Special Cases of Change of Variables . . . . . . . . . . . . . . . . . . . . . 99
4.3.1 Lebesgue Integrals on R . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.3.2 Linear Transformations on Rn . . . . . . . . . . . . . . . . . . . . . 102
4.3.3 Differentiable Transformations on Rn . . . . . . . . . . . . . . . . . . 106
References 231
Index 235
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Preface
The idea for a reference book on the mathematical foundations of quantitative finance has
been with me throughout my professional and academic careers in this field, but the com-
mitment to finally write it didn’t materialize until completing my introductory quantitative
finance book in 2010.
My original academic studies were in pure mathematics in a field of mathematical anal-
ysis, and neither applications generally nor finance in particular were then even on my
mind. But on completion of my degree, I decided to temporarily investigate a career in ap-
plied math, becoming an actuary, and in short order became enamored with mathematical
applications in finance.
One of my first inquiries was into better understanding yield curve risk management, ulti-
mately introducing the notion of partial durations and related immunization strategies. This
experience led me to recognize the power of greater precision in the mathematical specifica-
tion and solution of even an age-old problem. From there my commitment to mathematical
finance was complete, and my temporary investigation into this field became permanent.
In my personal studies, I found that there were a great many books in finance that
focused on markets, instruments, models, and strategies, which typically provided an in-
formal acknowledgment of the background mathematics. There were also many books on
mathematical finance focusing on more advanced mathematical models and methods, and
typically written at a level of mathematical sophistication requiring a reader to have signif-
icant formal training and the time and motivation to derive omitted details.
The challenge of acquiring expertise is compounded by the fact that the field of quanti-
tative finance utilizes advanced mathematical theories and models from a number of fields.
While there are many good references on any of these topics, most are again written at
a level beyond many students, practitioners and even researchers of quantitative finance.
Such books develop materials with an eye to comprehensiveness in the given subject matter,
rather than with an eye toward efficiently curating and developing the theories needed for
applications in quantitative finance.
Thus the overriding goal I have for this collection of books is to provide a complete and
detailed development of the many foundational mathematical theories and results one finds
referenced in popular resources in finance and quantitative finance. The included topics
have been curated from a vast mathematics and finance literature for the express purpose
of supporting applications in quantitative finance.
I originally budgeted 700 pages per book, in two volumes. It soon became obvious
this was too limiting, and two volumes ultimately turned into ten. In the end, each book
was dedicated to a specific area of mathematics or probability theory, with a variety of
applications to finance that are relevant to the needs of financial mathematicians.
My target readers are students, practitioners, and researchers in finance who are quanti-
tatively literate and recognize the need for the materials and formal developments presented.
My hope is that the approach taken in these books will motivate readers to navigate these
details and master these materials.
Most importantly for a reference work, all 10 volumes are extensively self-referenced.
The reader can enter the collection at any point of interest, and then using the references
xi
xii Preface
cited, work backward to prior books to fill in needed details. This approach also works for
a course on a given volume’s subject matter, with earlier books used for reference, and for
both course-based and self-study approaches to sequential studies.
The reader will find that the developments herein are presented at a much greater level
of detail than most advanced quantitative finance books. Such developments are of necessity
typically longer, more meticulously reasoned, and therefore can be more demanding on the
reader. Thus before committing to a detailed line-by-line study of a given result, it can be
more efficient to first scan the derivation once or twice to better understand the overall logic
flow.
I hope the scope of the materials, and the additional details presented, will support your
journey to better understanding.
I am grateful for the support of my family: Lisa, Michael, David, and Jeffrey, as well as
the support of friends and colleagues at Brandeis International Business School.
Robert R. Reitano
Brandeis International Business School
Author
xiii
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Introduction
xv
xvi Introduction
the integrals of nonnegative functions, including Fatou’s lemma and Lebesgue’s monotone
convergence theorem, which are then instrumental in the development of the properties of
such integrals. As an application of this theory, a Chapter I.7 result is generalized, address-
ing countable additivity of the set function defined on measurable rectangles of product
measure spaces.
This integration theory is then applied to derive integrals of general measurable func-
tions and their properties, along with the integration to the limit results of Beppo-Levi’s
theorem and Lebesgue’s dominated convergence theorem. The bounded convergence theo-
rem is then derived, as is an integration to the limit result not seen in Book III, called the
uniform integrability convergence theorem. The Leibniz integral rule on the derivatives of
parametrically defined integrals is then studied, generalizing the result of the Riemann the-
ory of Book III. The chapter ends with a discussion of when the Lebesgue-Stieltjes integrals
of this chapter agree with the Riemann-Stieltjes integrals of Book III.
Chapter 4 develops various results on change of variables in Lebesgue-Stieltjes integrals.
The first investigation is fundamental in probability theory and addresses integrals whose
measures are defined by integrals of measurable functions with respect to other measures.
In probability theory, for example, this measure would be the Borel measure on R induced
by the density function of a given random variable.
A more general realization of this idea is studied next, related to measurable transfor-
mations and their induced measures on range spaces. Integrals on the domain and range
spaces can then be related using this transformation and related measures. Special cases
of transformations are then studied in detail, beginning with linear transformations on Rn ,
and then turning to continuously differentiable transformations.
Integrals on product spaces are studied in Chapter 5. The key question is, echoing a Rie-
mann result of Book III, when can a multiple integral be evaluated using so-called iterated
integrals, which integrate one variable at a time? Such results prove to be related to the
product space sigma-algebra used, whereby the choices contrasted are the complete sigma
algebra and the smallest sigma algebra that contains the defining algebra. In large measure,
the results look quite similar, but there are details related to the need to qualify certain
statements with “almost everywhere.” The fundamental results here are then Fubini’s the-
orem, applicable to integrable functions, and Tonelli’s theorem, applicable to nonnegative
measurable functions. Examples from earlier books are then used to illustrate the theory,
though the next two chapters find deeper applications.
In Chapter 6, the first application of the Chapter 5 theory is to Lebesgue-Stieltjes
integration by parts. For this, the notion of a “signed measure” is introduced and seen to
possess all the properties of a measure other than nonnegativity. The integration theory is
then developed for such measures when induced by functions of bounded variation, echoing
the Riemann-Stieltjes work of Book III. The second application of the Chapter V theory
is to the study of the integrability of the convolution of integrable functions, proving that
such functions are indeed integrable.
The theory of Fourier transforms is developed in Chapter 7, a theory that will have
its most important applications in Book VI in the guise of characteristic functions. After
introducing the notion of the integral of complex-valued functions, the Fourier transform of
integrable functions and finite Borel measures is defined and various properties developed. In
particular, there are key connections between the decay at infinity of a function (or measure)
and the differentiability of its Fourier transform. Conversely, there are connections between
the existence and integrability of the derivatives of a function and the rate of decay at
infinity of its Fourier transform.
Fourier inversion is then studied, whereby one recovers the measure from its Fourier
transform, or, recovers the integrable function in the case where this transform is integrable.
Finally, a continuity theory is studied, which addresses the connection between the weak
Introduction xvii
convergence of probability measures and the pointwise convergence of their Fourier trans-
forms. An application to Poisson’s limit theorem is made, while the more powerful applica-
tions are deferred to Book VI.
Chapter 8 investigates general measure relationships and various decompositions of mea-
sures vis-a-vis other measures. The chapter begins with an example of the decomposition of
a Borel measure on R into measures that have characteristic properties relative to Lebesgue
measure. These characteristic properties are then generalized, whereby given a measure µ
we identify what it means for other measures to be absolutely continuous, or, mutually sin-
gular, relative to µ. To generalize the Borel decomposition, the signed measures introduced
in Chapter 6 are now studied in some detail. Various results are developed, such as the
Hahn decomposition, which address the positive and negative sets underlying a signed mea-
sure, and the Jordan decomposition, which decomposes a signed measure into a difference
of measures.
Perhaps the most famous and useful result on relationships between measures is the
Radon-Nikodým theorem. It states that if a measure υ is absolutely continuous with re-
spect to a measure µ, then υ is given by the µ-integral of a measurable function. This key
result will have an important application in Chapter 9, and then in the stochastic pro-
cess studies of Books VII–IX. The chapter ends with the Lebesgue decomposition theorem,
which generalizes the chapter’s Borel example to σ-finite measures.
The final Chapter 9 investigates Banach spaces, adding to their introduction in Book
III by studying the so-called Lp spaces of variously integrable or bounded measurable func-
tions, and their properties. Bounded linear functionals on Lp spaces are investigated and
characterized by the Riesz representation theorem, which is proved with the aid of the
Radon-Nikodým theorem. The special space of L2 , which is a Hilbert space, is addressed.
I hope this book and the other books in the collection serve you well.
Notation 0.1 (Referencing within FQF Series) To simplify the referencing of results
from other books in this series, we use the following convention.
A reference to “Proposition I.3.33” is a reference to Proposition 3.33 of Book I, while
“Chapter III.4” is a reference to Chapter 4 of Book III, and “II.(8.5)” is a reference to
formula (8.5) of Book II, and so forth.
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http://taylorandfrancis.com
1
Measure Spaces
This chapter summarizes some of the more important results from Book I on the various
constructions of a measure space (X, σ(X), µ). The importance of such constructions is
twofold:
• The requirements for a measure space (X, σ(X), µ) are quite demanding and, in fact,
so demanding that it is not immediately clear that such objects even exist.
• Given general existence, a measure space with particular properties will often be re-
quired. What if any restrictions are needed on these properties to ensure existence? For
example, can we create a measure on R so that the measure of any interval [a, b] is given
by f (b) − f (a) for a given function f ?
For existence, the space X is generally a collection of points, sometimes with other
special properties, but any set X can in theory be used. By Definitions I.2.1 and I.2.5, the
sigma algebra σ(X) is a collection of subsets of X with the following properties:
Definition 1.1 (Algebra of sets, sigma algebra) A collection A of sets from a space
X is called a algebra of sets on X:
S
1. If A ∈ A and B ∈ A, then the union A B ∈ A where:
[
A B ≡ {x|x ∈ A or x ∈ B}.
2. If A ∈ A, then A
e ∈ A, where the complement of A is defined:
e ≡ {x ∈ X|x ∈
A / A}.
The complement of A is also denoted Ac .
A collection σ(X) of sets from a space X is called a sigma algebra on X if σ(X) is
an algebra of sets, and:
S∞
3. If {Ai }∞
i=1 ⊂ σ(X), then i=1 Ai ∈ σ(X).
Thus algebras are closed under finite unions and complementation, and also finite inter-
sections by De Morgan’s laws of Exercise I.2.2. A sigma algebra is closed under countable
unions, and thus again countable intersections. In addition, algebras, and hence also sigma
algebras,
Smust contain X and the empty set ∅. For example, A ∈ A implies A e ∈ A and thus
both A A e = X ∈ A and A T A e = ∅ ∈ A.
The existence of algebras and sigma algebras can be demonstrated starting with any
collection of sets. Such a collection then obtains both an algebra and a sigma algebra by
defining A as the smallest algebra that contains this collection, and σ(X) as the smallest
such sigma algebra. This construction works because by Proposition I.2.8, the intersection
of any collection of algebras is an algebra, and similarly for sigma algebras. To ensure that
we do not have a vacuous intersection in this construction, note that the power set P(X),
which contains all subsets of X, is one such algebra and sigma algebra.
An important example of such constructions are the Borel sigma algebras.
DOI: 10.1201/9781003264576-1 1
2 Measure Spaces
Example 1.2 (Borel sigma algebra) By Definition I.2.13, the Borel sigma algebra
n
B(R ) on Rn is the smallest sigma algebra that contains the open sets of Rn . More generally,
if X is a topological space (Definition 1.5), the Borel sigma algebra B(X) is defined as the
smallest sigma algebra on X that contains all the open sets as defined by the topology T
on X.
By De Morgan’s laws, Borel sigma algebras can also be defined as the smallest sigma
algebras that contain the closed sets, but the above formulation is conventional.
Recall the definition of open and closed sets. For more background on open and closed
sets in various contexts, see Chapter 4 of Reitano (2010) or Section III.4.3.2, and also
Dugundji (1970) or Gemignani (1967).
Definition 1.4 (Open sets in R, Rn , and metric X) A set E ⊂ R is called open if for
any x ∈ E there is an open interval containing x that is also contained in E. In other words,
there exists 1 , 2 > 0 so that (x − 1 , x + 2 ) ⊂ E. There is no loss of generality by requiring
1 = 2 .
A set E ⊂ Rn is called open if for any x ∈ E there is an open ball Br (x) about x of
radius r > 0:
Br (x) ≡ {y| |x − y| < r},
so that Br (x) ⊂ E. Here |x − y| denotes the standard metric on Rn :
hXn i1/2
2
|x − y| ≡ (xi − yi ) .
i=1
More generally, if X is a metric space with metric d, a set E ⊂ X is called open if for
any x ∈ E there is an open ball Br (x) about x of radius r > 0:
so that Br (x) ⊂ E.
In all cases, a set F is called closed if Fe, the complement of F, is open.
The above notions of an open set reflect the natural metric |·| on R and Rn , where
d(x, y) ≡ |x − y| , or more generally a metric d on a space X. For more on metrics, see the
above references and Section III.4.3.1.
Open sets can also be defined without metrics, and this notion allows one to then define
a continuous function. See Proposition 2.10.
1. ∅, X ∈ T ;
2. If {Aα }α∈I S ⊂ T , where the index set I is arbitrary (finite, countably infinite, uncount-
able), then α∈I Aα ∈ T ;
Tn
3. If {Ai }ni=1 ⊂ T , then i=1 Ai ∈ T .
Measure Spaces 3
Exercise 1.6 (Natural topology on Rn ) Prove that the collection of open sets on R, as
defined in Definition 1.4, is a topology on R, and then generalize to Rn . These topologies are
often referred to as the natural topologies on R and Rn , and also the Euclidean topolo-
gies. The same result is true for the metric space (X, d), and this is called the topology
induced by the metric d. Hint: Recall Definition III.4.39 for definitional properties of
metrics.
Returning to the existence question for a measure space (X, σ(X), µ), since it is relatively
easy to construct various (X, σ(X)) pairs, the real question of existence relates to the
existence of a measure µ on σ(X). Recalling Definition I.2.23:
Definition 1.7 (Measure space) Let X be a set and σ(X) a sigma algebra of subsets of
X. A measure µ on X is a nonnegative set function defined on σ(X), taking values in
+
the nonnegative extended real numbers R ≡ R+ ∪{∞}, and which satisfies the following
properties:
1. µ(∅) = 0;
2. Countable Additivity: If {Ai }∞i=1 is aT
countable collection of pairwise disjoint sets
in the sigma algebra σ(X), meaning Aj Ak = ∅ if j 6= k, then:
[∞ X∞
µ Ai = µ (Ai ) . (1.1)
i=1 i=1
In such a case, the sets in σ(X) are said to be measurable, and sometimes µ-
measurable, and the triplet (X, σ(X), µ) is called a measure space.
Definition 1.8 (Lebesgue, Borel measure spaces) For these special measure spaces, a
third requirement is added:
m(I) = |I| = b − a,
the length of I.
By the construction in Section I.7.6, this criterion
Qn generalizes for Lebesgue measure on
Rn , that for any measurable rectangle R ≡ i=1 hai , bi i , whether such intervals are
open, closed, or semi-closed:
Yn
m(R) = |R| = (bi − ai ) ,
i=1
the volume of R.
n
• For a Borel measure µ, X = Rn and σ(X) = B(R ), and Definitions I.5.1 and
I.8.4 add the requirement:
n
3’. For any compact set A ∈ B(R ), µ(A) < ∞.
As noted above, while it is relatively easy to construct a sigma algebra on given set, it
is by no means apparent how one then defines a measure on this sigma algebra.
4 Measure Spaces
Remark 1.9 (Additional properties of measures) A few comments on the above def-
initions:
• Lebesgue translation invariance: For Lebesgue measure, it follows from item 3 that
m is translation invariant on intervals:
where I + x ≡ {x + y|y ∈ I} for any x. This property is then satisfied for all Lebesgue
measurable sets on R by Propositions I.2.28 and I.2.35, and analogously generalizes to
Rn by the construction of Chapter I.7.
• Borel measures: The notion of a Borel measure is not standardized, and it is not
uncommon to see other restrictions added in place of item 3’, such as the inner and
outer regularity properties of Proposition I.5.29. Measure spaces that also satisfy item
3’ are often called Radon measures, after Johann Radon (1887–1956).
• Finite additivity: Setting Ai = ∅ for i > n in (1.1), it follows from this definition that
all measures also satisfy finite additivity for pairwise disjoint sets:
[n Xn
µ Ai = µ (Ai ) . (1.2)
i=1 i=1
• Monotonicity: Measures are monotonic, which means that if A ⊂ B with both mea-
surable, then:
µ (A) ≤ µ (B) . (1.3)
S
This follows from finite additivity since B = A (B − A) as a disjoint union.
The existence of measures is more readily justified when X has a finite or countable
number of points.
For uncountable sets X, the construction is of necessity more subtle and sometimes ob-
tains surprising results. In the next section, we review the special constructions of Lebesgue
and Borel measure spaces of Chapters I.2 and I.5. We then summarize the general construc-
tion theory of Chapter I.6, and in the following section, recall Book I applications of this
general framework to various special constructions.
Lebesgue and Borel Spaces on R 5
• Lebesgue: The collection of sets is the open intervals G ≡ {(a, b)}, and the measure of
I = (a, b), denoted |I| , is defined as interval length:
|I| = b − a. (1.5)
where each In is an open interval, and |In | denotes its interval length.
• Borel: The collection of sets is the right semi-closed intervals A0 ≡ {(a, b]}, and the
measure of I = (a, b], denoted |I|F , is defined as the F -length of the interval:
and then by translation invariance derived that all such sets must have the same outer
measure.
This construction completely crushes any hope that m∗ is countably additive. Indeed,
it follows that depending on whether m∗ (A0 ) = 0 or m∗ (A0 ) > 0:
X∞
m∗ (Aj ) ∈ {0, ∞}.
j=1
∗
But in no case can it be that m ([0, 1]) = 1.
∗
P∞As m∗ is countably subadditive by Proposition I.2.29, it follows that m∗ (A0 ) > 0 and
∗
j=1 m (Aj ) = ∞. It then follows from this that m is not even finitely additive. So, the
problem here is even more serious than it first appears.
We are left with two possibilities:
There is no hope for item 1, since any proposal will encounter the Vitali construction.
To pursue item 2, there are two conventional approaches to identify the sets in P (R) on
which m∗ is indeed a measure, and these are discussed in Section I.2.4. The approach used
there was one that generalizes well to other constructions.
In Definition I.2.33, the approach of Constantin Carathéodory (1873–1950) was used,
an approach he developed for the general theory of outer measures.
It was then proved in a series of results culminating in Proposition I.2.39 that ML (R)
is a sigma algebra that contains the intervals and hence the Borel sigma algebra B(R), and
that m∗ is a Lebesgue measure on this sigma algebra. Lebesgue measure m is then defined
on ML (R) by:
m ≡ m∗ ,
and (R, ML (R), m) is a Lebesgue measure space.
It was also proved that ML (R) is a complete sigma algebra, and thus (R, ML (R), m)
is a complete measure space. See Definition 1.16. As noted above, ML (R) contains the
intervals, and m agrees with the specification in (1.5) on this collection.
An important corollary of the definition of the outer measure m∗ in (1.6) is that Lebesgue
measurable sets can be approximated well with open sets, and various classes of sets defined
using open sets. See Propositions I.2.42 for approximations, and I.2.43 for regularity of
Lebesgue measure.
To create a Borel measure space, it thus makes sense to investigate if any such increasing
and right continuous function F (x) can be used to induce a Borel measure µF . Indeed, the
answer is in the affirmative. Initially defining the set function F -length on right semi-closed
intervals (a, b] by (1.7), which is now compelled by (1.10), a series of results extends this
set function.
The first extension is to a set function µA on the algebra A, constructed as the collection
of all finite disjoint unions of right semi-closed intervals. This extension is defined additively
by disjointness. In Proposition I.5.13, µA proves to be a measure on this algebra.
1. µ(∅) = 0,
2. Countable Additivity: If {Aj }∞
j=1 is a countable collection of pairwise disjoint sets
S∞
in A with j=1 Aj ∈ A, then:
[ ∞ X∞
µ Aj = µ (Aj ) . (1.11)
j=1 j=1
An outer measure µ∗A is introduced as in (1.8), and as in the Lebesgue case, we utilize
the Carathéodory criterion in Definition I.5.16:
It was proved in a series of results culminating in Proposition I.5.23 that MµF (R) is a
sigma algebra that contains the algebra A and hence the Borel sigma algebra B(R), and
that µ∗A is a Borel measure on this sigma algebra. The Borel measure µF is then defined on
MµF (R) by:
µF ≡ µ∗A ,
and R, MµF (R), µF is a Borel measure space.
It was also proved that MµF (R) is a complete sigma algebra, and thus
R, MµF (R), µF is a complete measure space. See Definition 1.16. Further, A ⊂ MµF (R)
as noted above, and µF agrees with the specification in (1.7) on A.
An important corollary of the definition of the outer measure µ∗A in (1.6) is that sets
in MµF (R) can be approximated well with various classes of sets defined using right semi-
closed intervals. See Proposition I.5.26 and Remark I.5.27 for approximations and why these
appear to differ structurally from the Lebesgue approximations, and Proposition I.5.29 for
regularity of Borel measures.
1. If A01 , A02 ∈ A0 , then A01 A02 ∈ A0 , and thus this holds by induction for all finite
T
intersections.
f0 = Sn A0 .
2. If A0 ∈ A0 , then there exists disjoint {A0j }nj=1 ⊂ A0 so that A j=1 j
2’. If A ∈ A then A
e ∈ A.
The collection {(a, b]} is a semi-algebra, and by Exercise I.6.10, every semi-algebra A0
generates an algebra A defined as the collection of all finite disjoint unions of A0 -sets. This
General Extension Theory 9
implies that if we can define a set function on such a semi-algebra A0 , that the extension to
a measure on A will follow additively as in the Borel case.
Next, the notion of an outer measure is introduced. This definition captured the key
properties of the above special definitions that supported the conclusion that the collection
of Carathéodory measurable sets is a sigma algebra, and that µ∗ restricted to this
collection is a measure.
Definition 1.15 (Outer measure) Given a set X, a set function µ∗ defined on the power
sigma algebra σ(P (X)) of all subsets of X is an outer measure if:
1. µ∗ (∅) = 0.
2. Monotonicity: For sets A ⊂ B :
µ∗ (A) ≤ µ∗ (B).
This definition looks nothing like the outer measure definitions in (1.6) and (1.8). But
it can be noted that the first results developed after the Book I introductions of m∗ and
µ∗A were to establish that these outer measures had the above properties. Further, these
properties were essential in the subsequent developments.
The results seen in the Borel development are then completely generalized in a series of
Chapter I.6 results, which worked backward starting with the Carathéodory extension
theorem 1 of Proposition I.6.2 and named for Constantin Carathéodory (1873–1950). It
asserts that outer measures always obtain complete measure spaces by the Carathéodory
criterion.
Definition 1.16 (Complete measure space) A measure space (X, σ(X), µ) is com-
plete if for any A ∈ σ(X) with µ(A) = 0, then B ∈ σ(X) for all B ⊂ A. It then follows by
monotonicity of measures that µ(B) = 0 for all such B.
Equivalently, if A, C ∈ σ(X) with µ(A) = µ(C), then B ∈ σ(X) for all C ⊂ B ⊂ A, and
then µ(B) = µ(A) for all such B.
µ∗ (E) = µ∗ (A ∩ E) + µ∗ (A
e ∩ E). (1.13)
and:
2. Countable additivity: If {An }∞ 0
j=1 ⊂ A is a disjoint countable collection of sets and
S∞ 0
j=1 An ∈ A , then: [ ∞ X∞
µ0 Aj = µ0 (Aj ).
j=1 j=1
The algebra A in 1.b can be the algebra generated by a given semi-algebra A0 or defined
independently of a semi-algebra.
2. “Free” Steps:
This framework was applied in Chapters I.7–I.9 to obtain measure spaces in three im-
portant contexts. While still requiring a certain amount of effort to obtain the required step,
both the reader and the author were likely equally pleased to not have to then explicitly
derive all the results obtained in the free steps.
Definition 1.23 (Product space and set function) Given measure spaces {(Xi , σ(Xi ),
µi )}ni=1 , the product space:
Yn
X= Xi ,
i=1
is defined as:
X = {x ≡ (x1 , x2 , ..., xn )|xi ∈ Xi }. (1.15)
A measurable rectangle in X is a set A:
Yn
A= Ai = {x ∈ X|xi ∈ Ai }, (1.16)
i=1
0
where Ai ∈ σ(Xi ). We denote the collection of measurable
Qn rectangles in X by A .
The product set function µ0 is defined on A = i=1 Ai ∈ A0 by:
Yn
µ0 (A) = µi (Ai ), (1.17)
i=1
Unsurprisingly given the notation, the collection A0 proves to be a semi-algebra, and the
set function µ0 can be extended additively to a set function µA on the associated algebra
A of finite disjoint unions of A0 -sets. The derivation that µA so defined is a measure on
this algebra is subtle. This is in part due to the complexity of A-sets, and also that the
necessary results must be derived with (1.17) and properties of the measures {µi }ni=1 .
While generalized in Section 3.2.4, the Book I proof of countable additivity of µA required
that {µi }ni=1 be σ-finite measures. This obtained σ-finiteness of µA on A and of the resulting
measure µX on the complete sigma algebra, there denoted σ(X). The product measure space
of Proposition I.7.20 is then denoted (X, σ(X), µX ) . Qn
As noted in Notation I.7.21, it is common to express µX = i=1 µi . This notation also
reflects the fact that µX Q is an extension of µA from A to σ(X) and thus extends µ0 from
n
A0 to σ(X). So, for A = i=1 Ai ∈ A0 , (1.17) can be expressed:
Yn
µX (A) = µi (Ai ). (1.18)
i=1
When {(Xi , σ(Xi ), µi )}ni=1 = {(R, MFi (R), µFi )}ni=1 are Borel measure spaces, which
n Qn
are sigma finite by item 30 of Definition 1.8,Q
the final measure space (Rn , M(R ), i=1 µFi )
n n n n
contains a Borel measure space (Rn , B(R ), i=1 µFi ) since B(R ) ⊂ M(R ), and µX proves
to be a Borel measure.
However, there are Borel measures on Rn other than these product measures, and this
is the subject we discuss next.
We say that Fµ is continuous from above if the above property is true for all x.
2. Fµ satisfies
Qnthe n-increasing condition if given any bounded right semi-closed rect-
angle A = i=1 (ai , bi ] : X
sgn(x)Fµ (x) ≥ 0. (1.20)
x
Each x = (x1 , ..., xn ) in the summation is one of the 2n vertices of A, so xi = ai or
xi = bi , and sgn(x) equals −1 if the number of ai -components of x is odd, and equals
+1 otherwise.
Given the insights of this study of general Borel measures, the Chapter I.6 extension
n
theory is then applied to investigate if a Borel measure space (Rn , B(R ), µF ) can be con-
structed from a continuous from above and n-increasing function F : Rn → R. Given such
F (x), we begin by defining the class of bounded right semi-closed rectangles:
n
n
Yn o
A0B ≡ A ∈ B(R )|A = (ai , bi ], with − ∞ < ai ≤ bi < ∞ ,
i=1
and on A0B define the set function µ0 as in (1.21). It can be checked that A0B is not a
semi-algebra (Hint: consider A),
e so this will need to be addressed later in the development.
It is then proved in Proposition I.8.13 that µ0 is finitely additive and countably subad-
ditive on A0B . Since A0B is not a semi-algebra, Carathéodory’s extension theorem 2 cannot
be directly applied. Instead, the set function µ∗F is defined on A ⊂ Rn by:
nX [ o
µ∗F (A) = inf µ0 (An ) | A ⊂ An , An ∈ A0B , (1.22)
n
Definition 1.27 (Infinite product space and set function) Given probability spaces:
where
Qn Aj(i) ∈ σ(Xj(i) ). The cylinder set in (1.24) is said to be defined by J and
0
i=1 j(i) , and the collection of cylinder sets in X is denoted by A .
A
0 0
Qn product set function µ0 is defined on A as follows. If A ∈ A is defined by J
The
and i=1 Aj(i) , then:
Yn
µ0 (A) = µj(i) (Aj(i) ). (1.25)
i=1
The above restriction to probability spaces stems from the need to have µ0 (A) in (1.25)
well defined. For example, if J = (1, 2, ..., n) then:
and so µn+1 (Xn+1 ) = 1 is derived unless one of these sets has infinite or zero measure.
As the notation suggests, A0 so defined is a semi-algebra. Further, µ0 extends additively
to µA on the associated algebra A, and µA proves to be finitely additive and countably
subadditive. For countable additivity, the algebra A needed to be enlarged to A+ , and all
Xi were then restricted to R.
Definition 1.28 (Product space; general cylinder sets: A+ ) Given probability spaces
∞
{(R,
Q∞ B(R), µi )}i=1 , where B(R) denotes the Borel sigma algebra, the product space R =
N
The cylinder set in (1.26) will be said to be defined by H and J, and the collection of
general cylinder sets in RN is denoted by A+ .
The cylinder set A can also be characterized in terms of the projection mapping:
Yn Yn
πJ ≡ π j(i) : RN → Rj(i) ,
i=1 i=1
by:
A = π −1
J (H). (1.27)
+
For A ∈ A defined by H and J, the product set function µ0 is defined by:
where µJ denotes the finite dimensional product space probability measure associated with
{(R, B(R), µj(i) )}ni=1 .
or
Ai+1 ⊂ Ai , for all i.
In the former case, we are interested in the measure of the union, and in the latter, the
measure of the intersection.
The properties identified in this proposition are often referred to in terms of the “conti-
nuity” of measures and understood in the following sense. Given a collection of measurable
sets {Bj }∞
j=1 , define Ai by:
Si
1. Ai = j=1 Bj , or,
Ti
2. Ai = j=1 Bj , where it is assumed that µ(A1 ) < ∞.
16 Measure Spaces
Proposition 1.29 (Continuity of all measures) Given the measure space (X, σ (X) , µ)
and {Ai }∞
i=1 ⊂ σ (X):
Proof. To prove item 1, note that Ai ⊂ Ai+1 implies that µ(Ai ) ≤ µ(Ai+1 ) by monotonicity
of µ. Define B1 = A1 , and for i ≥ 2, let Bi = Ai − Ai−1 . Then {Bi }∞
i=1 ⊂ σ (X) are disjoint
sets, and: [∞ [∞
Ai = Bi .
i=1 i=1
By countable additivity of µ:
[∞ X∞
µ Ai = µ(Bi )
i=1 i=1
Xi
= µ(A1 ) + lim µ(Aj − Aj−1 ).
i→∞ j=2
Since Aj−1 and Aj − Aj−1 are disjoint with union Aj , finite additivity assures that:
Ti
For item 2, j=1 Aj = Ai by the nesting property, while monotonicity and the assump-
tion that µ(A1 ) < ∞ yields for all i :
\
i
µ Aj = µ(Ai ) < µ(A1 ).
j=1
Again by monotonicity, {µ(Ai )}∞i=1 is a bounded, nonincreasing sequence, and thus has a
well-defined limit as i → ∞, which proves (1.30).
2
Measurable Functions
f −1 (A) ∈ σ(X).
n
An extended real-valued transformation f : X → R defined on the measure space
(X, σ(X), µ) is said to be measurable, etc., if f −1 (A) ∈ σ(X) for every Borel set A ∈
n
B(R ).
More generally, a mapping f (x) between measure spaces (X, σ(X), µX ) and (Y, σ(Y ), µY )
is measurable or σ(X)/σ(Y )-measurable, if for all A ∈ σ(Y ):
f −1 (A) ∈ σ(X).
If D ⊂ X and f (x) is defined on D, then the criterion for measurability is the same as
above, and thus of necessity, D = f −1 (R) ∈ σ(X).
n
Remark 2.2 (On A ∈ B(R) or A ∈ B(R )) With f (x) an extended real-valued function,
n
f : X → R or f : X → R , it may seem odd that the measurability criterion only addresses
Borel sets in R and Rn . But note that if f is measurable by the above definition, then:
f −1 (∞) = X − f −1 (R/Rn ),
Notation 2.3 (σ(X)-measurable) Given the variety of labels used above to declare mea-
surability, σ(X)-measurable is the most accurate for extended real-valued functions and
transformations because the criterion f −1 (A) ∈ σ(X) for all Borel A is a sigma algebra
restriction. Measurability has nothing to do with the measure µ since there can be many
DOI: 10.1201/9781003264576-2 17
18 Measurable Functions
measures defined on a sigma algebra, and these do not affect which functions are measur-
able and which are not. Nonetheless, the use of µ-measurable is fairly common and rarely
causes confusion when there is one sigma algebra on the space.
However, there will be instances in coming studies where we will encounter measure
spaces (X, σ i (X), µ) with various sigma-algebras σ i (X). In other words, the space X is
fixed as is the measure µ, but there can be various sigma algebras on which µ satisfies
the definition of measure. A simple example but a common one is when (X, σ(X), µ) is a
measure space and σ i (X) ⊂ σ(X) is a sigma subalgebra, then (X, σ i (X), µ) is again a
measure space. But there can also be multiple sigma algebras with no such inclusions.
In these situations, the notion of a measurable function can become ambiguous, as
can the notion of a µ-measurable function. Thus when there is more than one sigma
algebra on X, it is necessary to say that f is σ(X)-measurable, identifying the defining
sigma algebra.
When f (x) is a mapping between general measure spaces (X, σ(X), µX ) and
(Y, σ(Y ), µY ), we will always say that f is σ(X)/σ(Y )-measurable as noted above.
Although a σ(X)-measurable function or transformation could be called σ(X)/B(R)-
n
measurable, or σ(X)/B(R )-measurable, this level of formality is rarely needed.
Although measurability of f −1 (A) for all A in the range space sigma algebra is the
requirement, it is not necessary to verify this condition for all such A to establish measura-
bility. This was seen in Proposition I.3.4 for Lebesgue or Borel measurability, meaning where
the respective domain space was (R, ML (R) , m) or (R, B (R) , m). Then, measurability for
all y of f −1 ((−∞, y)) is equivalent to this statement for all f −1 ([y, ∞)), all f −1 ((y, ∞)), or
all f −1 ((−∞, y]). Using the same ideas, this is equivalent to measurability of all f −1 ((a, b)).
In any of these cases, Proposition I.3.26 obtains that this is equivalent to measurability of
f −1 (A) for all A ∈ B (R) .
Exercise 2.5 Generalize the prior paragraph to Lebesgue or Borel measurability Qn of transfor-
mations defined on (Rn , ML (Rn ) , m) or (Rn , B (Rn ) , m). Show that if f −1 ( i=1 (−∞, yi ))
is measurable forQall y = (y1 , ...,
Qnyn ), then this
Qnis equivalent toQmeasurability of all f −1 (A)
n n
for A defined as i=1 [yi , ∞), i=1 (yi , ∞), i=1 (−∞, yi ] or i=1 (ai , bi ).
What is clear from the Book I development and the results of Exercise 2.5 is that these
collections of sets, and there are many others, have the property that they generate the Borel
sigma algebras B (R) or B (Rn ) , the sigma algebras of the range spaces. This generalizes as
might be expected.
In the following result, we specify this special collection as A0 , which is our standard
notation for a semi-algebra. This result is true for any collection of sets that generate σ(Y ),
not just semi-algebras. But we use this notation because it will often be the case that there
is an apparent semi-algebra A0 that generates an algebra A, which in turn generates the
range space sigma algebra σ(Y ). For example, this applies when (Y, σ(Y ), µY ) is a measure
space created by the extension theory of the prior chapter.
The following result is also true for f : (X1 , d1 ) → (X2 , d2 ) where, recalling Exercise 1.6,
open sets in X1 and X2 are those induced by these metrics. Details are left as an exercise
in changing notation.
Conversely, assume f −1 (G) is open for all open G ⊂ R. Let x0 ∈ Rn be given and
y0 = f (x0 ) ∈ R. Choose any open set G ⊂ R that contains y0 , for example, we could
choose G = R. By definition of open, there exists > 0 so that B (y0 ) ⊂ G. By assump-
tion f −1 (B (y0 )) is open in Rn and contains x0 . Again by definition of open, there exists
Bδ (x0 ) ⊂ f −1 (B (y0 )) and thus f (Bδ (x0 )) ⊂ B (y0 ). This now translates to the − δ
definition for continuity and the proof is complete.
This result now provides an immediate extension of the notion of continuity to functions
on topological spaces.
It now follows from this characterization of continuity that continuous functions are
σ(X)-measurable.
Proposition 2.12 (Continuous ⇒ µ-measurable) Given a measure space (X, σ(X), µ),
assume that X is also a topological space and that σ(X) contains the open sets of X, and
hence contains the Borel sigma algebra B(X). If f : X → R is continuous, then f is σ(X)-
measurable.
Proof. Consider the open set G ≡ (a, b). Since continuous, f −1 ((a, b)) is open in X, and
thus by definition:
f −1 ((a, b)) ∈ B(X) ⊂ σ(X).
The proof is complete by Proposition 2.7.
The next result summarizes that measurability is preserved under simple arithmetic
operations. We restrict to real-valued functions with range in R. The reader is referred to
Remark I.3.34 for a discussion on generalizing to extended real-valued functions with range
in R. Furthermore, items 1 and 2 remain true for measurable real-valued transformations
and this is left as an exercise.
Proposition 2.13 Let f (x) and g(x) be real-valued σ(X)-measurable functions defined on
a measure space (X, σ(X), µ), and let a, b ∈ R. Then, the following are σ(X)-measurable:
1. af (x) + b,
2. f (x) ± g(x),
3. f (x)g(x),
4. f (x)/g(x) on {x|g(x) 6= 0}.
Proof. To simplify notation, the set {x|f (x) < r} is denoted by {f (x) < r}, and so forth.
Also, by Proposition 2.7, it is sufficient to prove that h−1 (A) ∈ σ(X) for any collection of
sets that generates B (R) , where h(x) denotes any function under consideration.
1. If a = 0, the function g(x) = b is σ(X)-measurable since g −1 (A) ∈ {∅, X} for all A. For
a > 0,
{af (x) + b < y} = {f (x) < (y − b)/a},
which is measurable since f (x) is σ(X)-measurable. A similar result applies to a < 0.
2. Consider the sum since then by part 1, −g(x) is measurable and this implies the result
for f (x) − g(x). For rational r, if f (x) < r and g(x) < y − r, then f (x) + g(x) < y.
Taking a union over all rational r:
[ h \ i
{f (x) < r} {g(x) < y − r} ⊂ {f (x) + g(x) < y}.
r
On the other hand, if f (x) + g(x) < y then f (x) < y − g(x), and by density of the
rationals, there exists rational r so that f (x) < r < y − g(x). This implies f (x) < r and
g(x) < y − r.
Hence, [ h \ i
{f (x) + g(x) < y} = {f (x) < r} {g(x) < y − r} ,
r
3. First note that both f 2 (x) and g 2 (x) are measurable. For f 2 (x), for example:
√ T √
{f (x) < y} {f (x) > − y}, y ≥ 0,
{f 2 (x) < y} =
∅, y < 0.
• y > 0: [
{1/g(x) < y} = {g(x) > 1/y} {g(x) < 0}.
• y = 0:
{1/g(x) < 0} = {g(x) < 0}.
• y < 0: \
{1/g(x) < y} = {g(x) > 1/y} {g(x) < 0}.
Thus 1/g(x) is measurable as is f (x)/g(x) by item 3.
The next result is a good example of when a measurability conclusion requires complete-
ness of the measure space (X, σ(X), µ). Recall that when f (x) = g(x), except on a set of
µ-measure 0, this is often written as f (x) = g(x) µ-a.e., and read, “µ almost everywhere.”
The first set is a subset of a set of µ-measure zero and is hence σ(X)-measurable by com-
pleteness, wheras the second set is the intersection of measurable E,
e the complement of E,
and σ(X)-measurable {x|f (x) < y}.
{fn (x)}∞
−∞, n=1 unbounded below,
inf n fn (x) = (2.4)
max{y|y ≤ fn (x) all n}, {fn (x)}∞
n=1 bounded below.
{fn (x)}∞
∞, n=1 unbounded above,
supn fn (x) = (2.5)
min{y|y ≥ fn (x) all n}, {fn (x)}∞
n=1 bounded above.
Definition 2.16 (Limits inferior/superior) Given a sequence of functions {fn (x)}∞ n=1 ,
the limit inferior and
T∞limit superior of the sequence are defined pointwise as follows.
For each x ∈ D ≡ n=1 Dmn{fn }:
When clear from the context, the subscript n → ∞ is often dropped from the lim inf and
lim sup notation.
Notation 2.17 The limit superior of a function sequence is alternatively denoted limfn (x),
and the limit inferior denoted limfn (x), but we will use the above notation throughout these
books.
Exercise 2.18 (The lim in lim inf and lim sup) From Definition 2.16, it may not be ap-
parent where the notion of limit appears. Prove that:
In other words, the limit inferior is the limit of infima, wheras the limit superior is the limit
of suprema. Hint: Consider how inf k≥n fk (x) and supk≥n fk (x) vary with n.
As anticipated, these limiting functions are σ(X)-measurable when the functions in the
sequence are σ(X)-measurable. For item 7, we recall Corollary I.3.46 that lim fn (x) exists
at x if and only if:
−∞ < lim inf fn (x) = lim sup fn (x) < ∞. (2.8)
Proposition 2.19 (Measurability of functions derived from {fn (x)}∞ n=1 ) Given a se-
quence of σ(X)-measurable functions {fn (x)}∞ n=1 defined on measurable domains
{Dn }∞
n=1 of the measure space
T∞ (X, σ(X), µ), the following functions are also σ(X)-
measurable as defined on D ≡ n=1 Dn :
24 Measurable Functions
Proof. By Proposition 2.7, it is sufficient to prove that h−1 (A) ∈ σ(X) for any collection
of sets that generate B (R) where h(x) denotes any function under consideration.
Item 1 follows from item 3, and 2 from 4, by defining fn (x) = fN (x) for n ≥ N.
For item 3, if h(x) is defined by h(x) = inf fn (x), then by (2.4):
Thus: \∞
{x|h(x) > y} = {x|fn (x) > y},
n=1
and is measurable as the intersection of measurable sets.
Similarly, with g(x) = sup fn (x):
\∞
{x|g(x) < y} = {x|fn (x) < y},
n=1
Corollary 2.20 (Measurability on complete (X, σ(X), µ)) Given a sequence of σ(X)-
measurable functions {fn (x)}∞ ∞
n=1 defined on σ(X)-measurable domains {Dn }n=1 of the com-
plete measure space (X, σ(X), µ), if h(x) denotes any of the functions identified in Propo-
sition 2.19, and g(x) = h(x) µ-a.e., then g(x) is σ(X)-measurable.
Proof. This is Proposition 2.14.
Proposition 2.21 Given (X, σ(X), µ), let {fn (x)}∞ n=1 be a sequence of real-valued σ(X)-
measurable functions defined on a measurable set D with µ(D) < ∞, and let f (x) be a
real-valued function so that fn (x) → f (x) pointwise for x ∈ D. Then given > 0 and δ > 0,
there is a measurable set A ⊂ D and an integer N , so that µ(A) < δ, and for all x ∈ D − A
and all n ≥ N :
|fn (x) − f (x)| < .
Proof. Given > 0, define:
Corollary 2.22 If (X, σ(X), µ) is complete, the conclusion of the above proposition re-
mains valid if fn (x) → f (x) for each x ∈ D outside a set of µ-measure 0.
Proof. Everything in the above proof remains the same, except that we can now only con-
clude that for every xT∈ D outside an exceptional set of measure 0, that there exists DN with
x∈ / DN , and hence N DN equals this set of measure 0. But then, limN →∞ µ[ DN ] → 0
again by Proposition 1.29, and the proof follows as above, with a final application of Corol-
lary 2.20.
This proposition does not imply that fn (x) converges uniformly to f (x) on D − A
because the set A depends on the given and δ. This result is close to but not equivalent to
Littlewood’s third principle of Chapter I.4, named for J. E. Littlewood (1885–1977).
To improve this result to Littlewood’s conclusion of “nearly uniform convergence,” it
must be shown that A can be chosen so that fn (x) → f (x) uniformly on D − A. That is,
we need to find a fixed set A with µ(A) < δ, so that for any > 0, there is an N, such that
|fn (x) − f (x)| < for all x ∈ D − A and all n ≥ N. See the introduction to Chapter I.4 for
more on Littlewood’s principles.
This next result formalizes Littlewood’s third principle and is known as Egorov’s the-
orem, named for Dmitri Fyodorovich Egorov (1869–1931), and sometimes phonetically
translated to Egoroff. It is also known as the Severini–Egorov theorem in recognition
of the somewhat earlier and independent proof by Carlo Severini (1872–1951).
Proposition 2.23 (Severini-Egorov theorem ) Given (X, σ(X), µ), let {fn (x)}∞ n=1 be
a sequence of σ(X)-measurable functions defined on a measurable set D with µ(D) < ∞,
and let f (x) be a σ(X)-measurable function so that fn (x) → f (x) pointwise for x ∈ D.
Then given δ > 0, there is a measurable set A ⊂ D with µ(A) < δ, so that fn (x) → f (x)
uniformly on D − A.
That is, for > 0, there is an N , so that |fn (x) − f (x)| < for all x ∈ D − A and
n ≥ N.
26 Measurable Functions
Proof. Given δ > 0, for each m define m = 1/m and δ m = δ/2m and apply Proposition
2.21. The result is a set Am with µ(Am ) < δ m , and
S∞an integer Nm , so that |fn (x) − f (x)| <
m for nP≥ Nm and all x ∈ D − Am . With A ≡ m=1 Am , countable subadditivity obtains
∞
µ(A) ≤ m=1 µ(Am ) = δ. We now claim that fn (x) → f (x) uniformly on D − A.
Given there is an m so that m < , and hence an Nm so that |fn (x) − f (x)| < m <
for n ≥ Nm and all x ∈ D − Am . But then this statement is also true for x ∈ D − A since
Am ⊂ A.
Corollary 2.24 (Severini–Egorov theorem) If (X, σ(X), µ) is complete, the above re-
sult remains valid if fn (x) → f (x) µ-a.e. for x ∈ D.
Proof. Left as an exercise.
Remark 2.25 (On µ(D) < ∞) To perhaps state the obvious, the above results apply with-
out the explicit need for the restriction of µ(D) < ∞ in finite measure spaces, and in par-
ticular, in probability spaces. In such a space, we can conclude that pointwise convergence
on any measurable set assures nearly uniform convergence.
The accents of the Homeric hexameter are the soft rustle of a leaf
in the midday sun, the rhythm of matter; but the “Stabreim” likes
“potential energy” in the world-pictures of modern physics, creates a
tense restraint in the void without limits, distant night-storms above
the highest peaks. In its swaying indefiniteness all words and things
dissolve themselves—it is the dynamics, not the statics, of language.
The same applies to the grave rhythm of Media vita in morte sumus.
Here is heralded the colour of Rembrandt and the instrumentation of
Beethoven—here infinite solitude is felt as the home of the Faustian
soul. What is Valhalla? Unknown to the Germans of the Migrations
and even to the Merovingian Age, it was conceived by the nascent
Faustian soul. It was conceived, no doubt, under Classic-pagan and
Arabian-Christian impressions, for the antique and the sacred
writings, the ruins and mosaics and miniatures, the cults and rites
and dogmas of these past Cultures reached into the new life at all
points. And yet, this Valhalla is something beyond all sensible
actualities floating in remote, dim, Faustian regions. Olympus rests
on the homely Greek soil, the Paradise of the Fathers is a magic
garden somewhere in the Universe, but Valhalla is nowhere. Lost in
the limitless, it appears with its inharmonious gods and heroes the
supreme symbol of solitude. Siegfried, Parzeval, Tristan, Hamlet,
Faust are the loneliest heroes in all the Cultures. Read the wondrous
awakening of the inner life in Wolfram’s Parzeval. The longing for the
woods, the mysterious compassion, the ineffable sense of
forsakenness—it is all Faustian and only Faustian. Every one of us
knows it. The motive returns with all its profundity in the Easter
scene of Faust I.